Bearing pressure
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Bearing pressure is a particular case of
contact mechanics Contact mechanics is the study of the deformation of solids that touch each other at one or more points.Johnson, K. L, 1985, Contact mechanics, Cambridge University Press.Popov, Valentin L., 2010, ''Contact Mechanics and Friction. Physical Pri ...
often occurring in cases where a convex surface (male cylinder or sphere) contacts a concave surface (female cylinder or sphere:
bore Bore or Bores often refer to: *Boredom * Drill Relating to holes * Boring (manufacturing), a machining process that enlarges a hole ** Bore (engine), the diameter of a cylinder in a piston engine or a steam locomotive ** Bore (wind instruments), ...
or hemispherical cup). Excessive contact pressure can lead to a typical bearing failure such as a plastic deformation similar to
peening In metallurgy, peening is the process of working a metal's surface to improve its material properties, usually by mechanical means, such as hammer blows, by blasting with shot (shot peening), or focusing light (laser peening). Peening is norma ...
. This problem is also referred to as bearing resistance.EN 1993-1-8:2005 '' Eurocode 3: Design of steel structures - Part 1-8: Design of joints''


Hypotheses

A contact between a male part (convex) and a female part (concave) is considered when the radii of curvature are close to one another. There is no tightening and the joint slides with no friction therefore, the contact forces are
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
to the tangent of the contact surface. Moreover, bearing pressure is restricted to the case where the charge can be described by a radial
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
pointing towards the center of the joint.


Case of a cylinder-cylinder contact

In the case of a
revolute joint A revolute joint (also called pin joint or hinge joint) is a one- degree-of-freedom kinematic pair used frequently in mechanisms and machines. The joint constrains the motion of two bodies to pure rotation along a common axis. The joint does no ...
or of a
hinge joint A hinge joint (ginglymus or ginglymoid) is a bone joint in which the articular surfaces are molded to each other in such a manner as to permit motion only in one plane. According to one classification system they are said to be uniaxial (having ...
, there is a contact between a male cylinder and a female cylinder. The complexity depends on the situation, and three cases are distinguished: * the clearance is negligible: ** a) the parts are
rigid bodies In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
, ** b) the parts are elastic bodies; * c) the clearance cannot be ignored and the parts are elastic bodies. By "negligible clearance", H7/g6 fit is typically meant. The axes of the cylinders are along the ''z''-axis, and two external forces apply to the male cylinder: * a force \vec along the ''y''-axis, the load; * the action of the bore (contact pressure). The main concern is the contact pressure with the bore, which is uniformly distributed along the ''z''-axis. Notation: * ''D'' is the nominal diameter of both male and female cylinders; * ''L'' the guiding length.


Negligible clearance and rigid bodies

In this first modeling, the pressure is uniform. It is equal to: : P = \frac = \frac.


Negligible clearance and elastic bodies

If it is considered that the parts deform elastically, then the contact pressure is no longer uniform and transforms to a sinusoidal repartition: : ''P''(θ) = ''P''max⋅cos θ with : P_\mathrm = \frac \cdot \frac. This is a particular case of the following section (θ0 = π/2). The maximum pressure is 4/π ≃ 1.27 times bigger than the case of uniform pressure.


Clearance and elastic bodies

In cases where the clearance can not be neglected, the contact between the male part is no longer the whole half-cylinder surface but is limited to a 2θ0 angle. The pressure follows
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
: : ''P''(θ) = ''K''⋅δα(θ) where * ''K'' is a positive real number that represents the rigidity of the materials; * δ(θ) is the radial displacement of the contact point at the angle θ; * α is a coefficient that represents the behaviour of the material: ** α = 1 for metals (purely elastic behaviour), ** α > 1 for polymers (
viscoelastic In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
or
viscoplastic Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The in ...
behaviour). The pressure varies as: : ''A''⋅cos θ - ''B'' where ''A'' and ''B'' are positive real number. The maximum pressure is: : P_\mathrm = \frac \times \frac the angle θ0 is in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s. The rigidity coefficient ''K'' and the half contact angle θ0 can not be derived from the theory. They must be measured. For a given system — given diameters and materials —, thus for given ''K'' and clearance ''j'' values, it is possible to obtain a curve θ0 = ƒ(''F''/(''DL'')).


Case of a sphere-sphere contact

A sphere-sphere contact corresponds to a
spherical joint In an automobile, ball joints are spherical bearings that connect the control arms to the steering knuckles, and are used on virtually every automobile made. They bionically resemble the ball-and-socket joints found in most tetrapod animals. ...
(socket/ball), such as a ball jointed cylinder saddle. It can also describe the situation of bearing balls.


Case of uniform pressure

The case is similar as above: when the parts are considered as rigid bodies and the clearance can be neglected, then the pressure is supposed to be uniform. It can also be calculated considering the projected area: : P = \frac = \frac.


Case of a sinusoidal repartition of pressure

As in the case of cylinder-cylinder contact, when the parts are modeled as elastic bodies with a negligible clearance, then the pressure can be modeled with a sinusoidal repartition: : ''P''(θ, φ) = ''P''max⋅cos θ with : P = \frac \cdot \frac.


Hertz contact stress

When the clearance can not be neglected, it is then necessary to know the value of the half contact angle θ0 , which can not be determined in a simple way and must be measured. When this value is not available, the Hertz contact theory can be used. The Hertz theory is normally only valid when the surfaces can not conform, or in other terms, can not fit each other by elastic deformation; one surface must be convex, the other one must be also convex plane. This is not the case here, as the outer cylinder is concave, so the results must be considered with great care. The approximation is only valid when the inner radius of the container ''R''1 is far greater than the outer radius of the content ''R''2, in which case the surface container is then seen as flat by the content. However, in all cases, the pressure that is calculated with the Hertz theory is greater than the actual pressure (because the contact surface of the model is smaller than the real contact surface), which affords designers with a safety margin for their design. In this theory, the radius of the female part (concave) is negative. A relative diameter of curvature is defined: : \frac = \frac + \frac where ''d''1 is the diameter of the female part (negative) and ''d''2 is the diameter of the male part (positive). An equivalent module of elasticity is also defined: : \frac = \frac + \frac where ν''i'' is the
Poisson's ratio In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
of the material of the part ''i'' and ''Ei'' its
Young's modulus Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
. For a cylinder-cylinder contact, the width of the contact surface is: : b = \left ( \frac \cdot \frac \right )^ and the maximal pressure is in the middle: : P_\max = \frac = \sqrt. In case of a sphere-sphere contact, the contact surface is a disk whose radius is: : a = \left ( \frac \cdot \frac \right )^ and the maximal pressure is in the middle: : P_\max = \frac = \frac \sqrt /math>.


Applications


Bolt used as a stop

In a bolted connection, the role of the bolts is normally to press one parts on the other; the adherence (
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
) is opposed to the tangent forces and prevents the parts from sliding apart. In some cases however, the adherence is not sufficient. The bolts then play the role of stops: the screws endure
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
whereas the hole endure bearing pressure. In good design practice, the threaded part of the screw should be small and only the smooth part should be in contact with the plates; in the case of a shoulder screw, the clearance between the screw and the hole is very small ( a case of rigid bodies with negligible clearance). If the acceptable pressure limit ''P''lim of the material is known, the thickness ''t'' of the part and the diameter ''d'' of the screw, then the maximum acceptable tangent force for one bolt ''F''b, Rd (design bearing resistance per bolt) is: : ''F''b, Rd = ''P''lim × ''d'' × ''t''. In this case, the acceptable pressure limit is calculated from the ultimate tensile stress ''f''u and factors of safety, according to the Eurocode 3 standard. In the case of two plates with a single overlap and one row of bolts, the formula is: : ''P''lim = 1.5 × ''f''uM2 where * γM2 = 1.25: partial safety factor. In more complex situations, the formula is: : ''P''lim = ''k''1 × α × ''f''uM2 where * ''k''1 and α are factors that take into account other failure modes than the bearing pressure overload; ''k''1 take into account the effects that are perpendicular to the tangent force, and α the effects along the force; * ''k''1 = min for end bolts,
''k''1 = min for inner bolts, ** ''e''2: edge distance from the centre of a fastener hole to the adjacent edge of the part, measured at right angles to the direction of load transfer, ** ''p''2: spacing measured perpendicular to the load transfer direction between adjacent lines of fasteners, ** ''d''0: diameter of the passthrough hole; * α = min, with ** ''e''1: end distance from the center of a fastener hole to the adjacent end of the part, measured in the direction of load transfer, ** ''p''1: spacing between centers of fasteners in the direction of load transfer, ** ''f''ub: specified ultimate tensile strength of the bolt. When the parts are in wood, the acceptable limit pressure is about 4 to 8.5 MPa.


Plain bearing

In
plain bearing A plain bearing, or more commonly sliding contact bearing and slide bearing (in railroading sometimes called a solid bearing, journal bearing, or friction bearing), is the simplest type of bearing, comprising just a bearing surface and no rolli ...
s, the
shaft Shaft may refer to: Rotating machine elements * Shaft (mechanical engineering), a rotating machine element used to transmit power * Line shaft, a power transmission system * Drive shaft, a shaft for transferring torque * Axle, a shaft around whi ...
is usually in contact with a bushing (sleeve or flanged) to reduce
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
. When the rotation is slow and the load is radial, the model of uniform pressure can be used (small deformations and clearance). The product of the bearing pressure times the circumferential sliding speed, called load factor PV, is an estimation of the resistance capacity of the material against the frictional heating.


References


Bibliography

* ublin 1992 * hevalier 2004 * anchon 2001 * anchon 2011 * CM 2000 * G 2003 {{Authority control Bearings (mechanical) Mechanical engineering Solid mechanics